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什么是博弈

通俗地讲， 博弈就是指游戏中的一种选择策略的研究">
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                    <h1 class="description center-align post-title">博弈论 | 三姬分金与囚徒困境</h1>
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                <h2 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h2><p>昨天在刷李永乐老师视频的时候看到了一个“三姬分金”的问题，然后引到了博弈论。说实话，从这个老师的课程中深深的感受到了知识的魅力。就想着写一个专栏，学习一下博弈论的有关知识。</p>
<p><strong>什么是博弈</strong></p>
<blockquote>
<p>通俗地讲， 博弈就是指游戏中的一种选择策略的研究。 博弈的英文为“game”， 我们一般将它翻译成“游戏”。 而在英语中， “game”的意义不同于汉语中的游戏， 它是人们遵循一定规则的活动， 进行活动的人的目的是让自己“赢”。 我们在和对手竞赛或游戏的时候怎样使自己赢呢？ 这不但要考虑自己的策略， 还要考虑其他人的选择。 生活中博弈的案例很多， 只要涉及人群的互动以及选择决策， 就有博弈。<br>一个人做选择时必须考虑其他人的选择或是事务的变化， 而其他人做选择时也会考虑此人的选择。 此人的结果——博弈论称之为支付， 不仅取决于他的行动选择——博弈论称之为策略选择， 同时取决于其他人的策略选择。 这样， 此人和其他人或事务就构成一个博弈。</p>
</blockquote>
<p><img src="https://img-blog.csdnimg.cn/20200411201405346.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzA5MjIzMg==,size_16,color_FFFFFF,t_70#pic_center" alt="博弈"></p>
<h2 id="三姬分金问题"><a href="#三姬分金问题" class="headerlink" title="三姬分金问题"></a>三姬分金问题</h2><h3 id="背景"><a href="#背景" class="headerlink" title="背景"></a>背景</h3><blockquote>
<p>当年，韩非子到一个大将军处索要军饷，看见大将军的三个妃子在玩分金币的游戏。于是韩非子提议，玩点更刺激的。</p>
</blockquote>
<h3 id="游戏规则"><a href="#游戏规则" class="headerlink" title="游戏规则"></a>游戏规则</h3><blockquote>
<p>现有A、B、C三名美姬，分100枚金币。从A开始，每个人依次提议分配方案。如果提议赞同人数不超过50%，则提议人被处死。反之则提议通过。A、B、C的顺序通过抽签来决定。<br>此外，有两点前提：<br>1.每个人都是聪明的，即她知道每个提议带来的后果，并追求自己利益的最大化。<br>2.每个人都是邪恶的，在她追求自己利益最大化的同时，要尽量多的杀死其余美姬。</p>
</blockquote>
<h3 id="思维实验0"><a href="#思维实验0" class="headerlink" title="思维实验0"></a>思维实验0</h3><p>如果游戏按照规则进行，那么结果会是怎样？<br>（1）我们不妨先假设A的提议被B、C否决：<br>美姬A被处死，到美姬B提议，根据规则”赞同人数不超过50%则提议人被处死”，故无论美姬B提议什么，美姬C都不会同意（即使B提议金币全给C也是如此）。美姬B这么聪明当然也考虑到了这个问题。<br>（2）美姬A提议必然通过：<br>由（1）中分析，美姬A的提议必然会被B同意（B不同意就死），所以A可以肆无忌惮的提议分配方案，故A提出了（100,0,0）的分配方案，实现了自己利益最大化。</p>
<p><strong>在思维实验0中，美姬A具有先手优势。</strong></p>
<hr>
<h3 id="思维实验1"><a href="#思维实验1" class="headerlink" title="思维实验1"></a>思维实验1</h3><p>假设在美姬A前面加一个美姬X，其他规则不变，结果会如何？<br>（1）不妨先假设美姬X的提议被否决：<br>美姬X被处死，到美姬A提议，参考思维实验0可知，美姬A还是提出（100,0,0）的提议，且必定被通过。<br>（2）美姬X提出方案对B，C有利则会被通过：<br>美姬X考虑到如果由美姬A提议，则B,C什么都得不到。故美姬X决定拉拢美姬B,C。她提出了（98,0，1,1）的分配方案。美姬A肯定不同意（但是没有用），美姬B,C一看还能得到1枚金币，于是通过了美姬X的提议方案。<br><strong>在思维实验1中，美姬X具有先手优势，即领导群体，美姬B、C属于基层群体，是领导群体拉拢的对象。在有限次的实验中，拿到先手权意味着锁定胜局。而且只要条件规则稳定，结果是一定的，这样的局面叫做纳什均衡</strong></p>
<hr>
<h3 id="思维实验2"><a href="#思维实验2" class="headerlink" title="思维实验2"></a>思维实验2</h3><p><strong>共谋</strong></p>
<blockquote>
<p>在思维实验1的背景下，美姬A,B,C觉得不公平，还想实现利益最大化。于是A提议：否定美姬X的提议，A,B,C各分33枚金币，剩下一枚丢进海里。这个过程叫做”共谋”。</p>
</blockquote>
<p>那么在实验的过程中，如何保证在杀掉美姬X之后，美姬A信守承诺均分金币呢？<br>A,B,C之间需要制定一个契约，规定B,C帮助A杀掉X之后，A能信守承诺均分金币。此外，为了防止A撕毁契约，还需要执行和守护契约的团体，通过分权的方式使其不能串通。执行和守护契约的团体的建立权在B,C手中，即实现了”民主”。</p>
<hr>
<h3 id="总结"><a href="#总结" class="headerlink" title="总结"></a>总结</h3><p>三姬分金这个故事很好的解释了社会中的一些现象：古代统治者为何需要拉拢底层人民、为何大副都想当船长…</p>
<blockquote>
<p>《1984》里面写到：有史以来，大概自从新石器时代结束以来，世上就有三种人，即上等人、中等人、下等人。 这三种人的目标是完全不可调和的。上等人的目标是要保持他们的地位。中等人的目标是要同高等人交换地位。下等人的特点始终是，他们劳苦之余无暇旁顾，偶而才顾到日常生活以外的事，因此他们如果有目标的话，无非是取消一切差别，建立一个人人平等的社会。</p>
</blockquote>
<hr>
<h2 id="囚徒困境问题"><a href="#囚徒困境问题" class="headerlink" title="囚徒困境问题"></a>囚徒困境问题</h2><h3 id="问题简介"><a href="#问题简介" class="headerlink" title="问题简介"></a>问题简介</h3><p><a href="https://wiki.mbalib.com/wiki/%E5%9B%9A%E5%BE%92%E5%9B%B0%E5%A2%83" target="_blank" rel="noopener">囚徒困境</a>是博弈论的<a href="https://wiki.mbalib.com/wiki/%E9%9D%9E%E9%9B%B6%E5%92%8C%E5%8D%9A%E5%BC%88" target="_blank" rel="noopener">非零和博弈</a>中具代表性的例子，反映个人最佳选择并非团体最佳选择。虽然困境本身只属模型性质，但现实中的价格竞争、环境保护等方面，也会频繁出现类似情况。</p>
<hr>
<h3 id="问题描述"><a href="#问题描述" class="headerlink" title="问题描述"></a>问题描述</h3><p>1950年，由就职于兰德公司的梅里尔·弗拉德（Merrill Flood）和梅尔文·德雷希尔（Melvin Dresher）拟定出相关困境的理论，后来由顾问阿尔伯特·塔克（Albert Tucker）以囚徒方式阐述，并命名为“囚徒困境”。经典的囚徒困境如下：</p>
<blockquote>
<p>警方逮捕甲、乙两名嫌疑犯，但没有足够证据指控二人入罪。于是警方分开囚禁嫌疑犯，分别和二人见面，并向双方提供以下相同的选择：</p>
</blockquote>
<blockquote>
<p>若一人认罪并作证检举对方（相关术语称“背叛”对方），而对方保持沉默，此人将即时获释，沉默者将判监10年。<br><img src="https://img-blog.csdnimg.cn/20200411203310243.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzA5MjIzMg==,size_16,color_FFFFFF,t_70#pic_center" alt="一方背叛"></p>
</blockquote>
<blockquote>
<p>若二人都保持沉默（相关术语称互相“合作”），则二人同样判监一年。<br><img src="https://img-blog.csdnimg.cn/20200411203250576.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzA5MjIzMg==,size_16,color_FFFFFF,t_70#pic_center" alt="双方背叛/双方合作"></p>
</blockquote>
<blockquote>
<p>若二人都互相检举（互相“背叛”），则二人同样判监8年。</p>
</blockquote>
<p><strong>表格描述</strong></p>
<table>
<thead>
<tr>
<th>囚徒行为</th>
<th>甲沉默（合作）</th>
<th>甲认罪（背叛）</th>
</tr>
</thead>
<tbody><tr>
<td>乙沉默（合作）</td>
<td>二人同服刑1年</td>
<td>甲当即释放，乙服刑10年</td>
</tr>
<tr>
<td>乙认罪（背叛）</td>
<td>乙当即释放，甲服刑10年</td>
<td>二人同服刑8年</td>
</tr>
</tbody></table>
<p>如同博弈论的其他例证，囚徒困境假定每个参与者（即“囚徒”）都是利己的，即都寻求最大自身利益，而不关心另一参与者的利益。参与者某一策略所得利益，如果在任何情况下都比其他策略要低的话，此策略称为“严格劣势策略”，理性的参与者绝不会选择。另外，没有任何其他力量干预个人决策，参与者可完全按照自己意愿选择策略。</p>
<hr>
<h3 id="思维实验0-1"><a href="#思维实验0-1" class="headerlink" title="思维实验0"></a>思维实验0</h3><p>（1）从个体来看，囚徒的选择如下：</p>
<ul>
<li>如果对方选择沉默，背叛可以使我获释</li>
<li>如果对方选择背叛，背叛可以使我的刑期减少</li>
</ul>
<p><strong>即从个体角度出发，背叛无疑是最优选择。</strong> 这场博弈中唯一可能达到的纳什均衡，就是双方参与者都背叛对方，结果二人同样服刑8年。</p>
<p>（2）从团体角度来看，如果两个参与者都合作保持沉默，两人都只会被判刑一年，总体利益更高，结果也比两人背叛对方、判刑8年的情况较佳。但根据以上假设，二人均为理性的个人，且只追求自己个人利益。均衡状况会是两个囚徒都选择背叛，结果二人判决均比合作为高，总体利益较合作为低。这就是“困境”所在。例子漂亮地证明了：非零和博弈中，<a href="https://wiki.mbalib.com/wiki/%E5%B8%95%E7%B4%AF%E6%89%98%E6%9C%80%E4%BC%98" target="_blank" rel="noopener">帕累托最优</a>和<a href="https://wiki.mbalib.com/wiki/%E7%BA%B3%E4%BB%80%E5%9D%87%E8%A1%A1" target="_blank" rel="noopener">纳什均衡</a>是相冲突的。</p>
<hr>
<h3 id="思维实验1-1"><a href="#思维实验1-1" class="headerlink" title="思维实验1"></a>思维实验1</h3><p><strong>当囚徒困境多次重复发生</strong></p>
<blockquote>
<p>单次发生的囚徒困境，和多次重复的囚徒困境结果不会一样。在重复的囚徒困境中，博弈被反复地进行。因而每个参与者都有机会去“惩罚”另一个参与者前一回合的不合作行为。这时，合作可能会作为均衡的结果出现。欺骗的动机这时可能被受到惩罚的威胁所克服，从而可能导向一个较好的、合作的结果。作为反复接近无限的数量，纳什均衡趋向于帕累托最优。</p>
</blockquote>
<hr>
<h3 id="总结-1"><a href="#总结-1" class="headerlink" title="总结"></a>总结</h3><p><strong>囚徒困境主旨</strong></p>
<blockquote>
<p>囚徒困境的主旨为，囚徒们虽然彼此合作，坚决不坦白，可为全体带来最佳利益（累计和最少刑期），但在资讯不明的情况下，因为出卖同伙可为自己带来利益（缩短刑期）。同伙坦白可为他带来利益，因此彼此出卖虽违反最佳共同利益，反而是自己最大利益所在。但实际上，执法机构不可能设立如此情境来诱使所有囚徒招供，因为囚徒们必须考虑刑期以外之因素（出卖同伙会受到报复等），而无法完全以执法者所设立之利益（刑期）作考量。</p>
</blockquote>
<p><strong>囚徒困境的应用</strong></p>
<blockquote>
<p>许多行业的价格竞争都是典型的囚徒困境现象，每家企业都以对方为敌手，只关心自己的利益。在价格博弈中，只要以对方为敌手，那么不管对方的决策怎样，自己总是以为采取低价策略会占便宜，这就促使双方都采取低价策略。如可口可乐公司和百事可乐公司之间的竞争、各大航空公司之间的价格竞争等等。</p>
</blockquote>
<blockquote>
<p>在国内的家电大战中，虽然不是两个对手之间的博弈，但由于在众多对手当中每一方的市场份额都很大，每一个主体人的行为后果受对手行为的影响都很大，因此，其情景大概也是如此。如果清楚这种前景，双方勾结或合作起来，都制定比较高的价格，那么双方都可以因为避免价格大战而获得较高的利润。但是往往这些联盟处于利益驱动的“囚徒困境”，双赢也就成泡影。五花八门的价格联盟总是非常短命，道理就在这里。</p>
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